I would like to offer some collegial, but firm, pushback on part of this. Specifically the first point: it's a mischaracterization of one of the central pieces of the essay:

There’s a very large amount I disagree with.

I think people are overhyping the response. It came in two parts. The first involved claiming that children can’t tell imagination from reality, but all the psychological literature reviews cited by both parties agree that they can! They don’t think their imaginary friends have the same status as real people, and there aren’t comparable cases of 11 year olds believing in such things to the point of enduring harsh persecution.

The response to my response has indeed been overwhelmingly positive, and this makes me deeply nervous as a kind of natural instinct. I can totally believe that it is currently being overhyped. 

Regardless, this is not an accurate characterization of what I argue in the essay.  

I do not argue that the children were unable to tell imagination from reality:

On May 13th, 1917, Lucia engaged in precisely this kind of fantastical thinking, remaining aware all the while of the unique ontological category to which her constructed entity, the Lady, belonged. However, through the perspective of her religious faith, Lucia viewed this unique 'fantasy' category as containing real or perhaps even more-than-real entities -- entities which were not figures of her imagination, but of divine revelation.

Emphasis added.

I do not say that Lucia failed to distinguish between her imagination and reality — that she ‘saw’ an imaginary thing and thought, for some confused reason, that it had the same ontological status as her own mother or a nearby tree. I say that she ‘saw’ an imaginary thing, and classed it as something different. Something extant, but of a different nature than — again — her parents or her house. Some people in the (now voluminous) comment section mentioned an LDS phrase — “eye of faith,” the ability to “see” things in a way that is not limited to simple physical sight. We can imagine her thinking being something like that.

Insofar it makes sense to formally partition up a young child’s ontology, we can imagine Lucia’s looked something like this:

• Regular Things: ex. Mother, trees, Francisco, my shoes

• Special Things: ex. Our Lady, God, The Holy Spirit

• Imaginary Things: ex. Dragons under my bed, the knight Francisco says he keeps in his pantry

The first two kinds of things are out in the world — they are different! and are ‘perceived’ in different senses of that word! — but the last kind of thing is not, truly. (Does a mature religious person not adopt something like this very distinction? Is this so inconceivable?)

Perhaps this feels too speculative. “We can imagine,” “We could say.” Of course this is the only kind of thing the skeptic needs to be able to say in these discussions. But we can be more positive about this:

Sharon and Wooley’s research — yet again! — confirms my specific reading of things: Children distinguish between reality and imagination, yes: they also specifically distinguish between general imaginary entities and specific “event-specific” characters, like Santa Claus and — I do not mean to utter an impiety — Our Lady of Fatima. They have a lot of trouble getting things right with this second class of entity right:

By five years of age, children made systematic, if sometimes erroneous, category judgments for all but two entities (fairy and dragon). The majority of this age group correctly categorized all of the real entities as real and the monster and Superman as pretend (p < .05). However, the majority also incorrectly categorized the two event related fantastical entities—the Easter Bunny and Santa—as real (p < .001.)

— pg. 300, Sharon & Wooley, 2008

This trouble is compared of course to the standard performance of the children in the study, as well as some concerns they describe which also seem relevant:

Consistent with previous research (Samuels & Taylor, 1994; Taylor & Howell, 1973), young children in our study frequently failed to categorize entities correctly in terms o f their fantasy/reality status. Overall, they were correct less than half the time. Thus, in terms of assigning entities to the correct category, children of all ages in this study performed poorly. However, two additional analyses point to a more complex picture.

— pg. 305

What makes the picture more complex:

Big differences in basic fantastical temperament / further developmental differences between individual children. Also:

This pattern of responses underscores the importance of the larger social context in the development of children’s beliefs. The view of children as highly credulous has often failed to acknowledge the extent to which children are expected and encouraged to hold certain fantastical beliefs (cf. Woolley, 1997). It should be noted that Piaget, whose work contributed significantly to this view, made a strong distinction between children’s individual magical beliefs and the magical beliefs of society as a whole (Piaget, 1930). Yet these seemingly paradoxical trends—of both increasing accuracy and increasing error—can be traced to children’s developing knowledge o f and experience in the world.

— pg. 306

This article just isn’t it for making the case that it is implausible to posit that the children were behaving in a way that is — frankly — just extremely familiar to anyone with regular contact with children. It is this last point I want to dwell on: ‘implausibility’

The miracle-truther side has to grapple with this basic rhetorical difficulty:

People give them psycho-social explanations of the children’s behavior — attempting, in the spirit of intellectual engagement, to go beyond merely gesturing to the way that children act all the time. “Look at these studies you cite, they say that children behave in this particular way about imaginary things. That’s basically what we’re saying the children did.” The miracle-truther side responds that no, no — the children at Fatima behaved in different way — one that may look similar to the unscrupulous but is in fact materially different. “We can diagnose from testimony that it’s not imaginary play. Can you find me another example where children behaved just this way? It’s implausible!”

It’s implausible that children would act in a way contrary to the boundary conditions of some developmental research. Ok. (This is stipulating for now that they are presenting the conclusions of the studies appropriately — which as I say above I do not believe they are — but let’s roll with it right now.) The unspoken coda: it is much more plausible that: there is a super-powerful agent unlike anything we have ever seen, capable of, but not limited to:

• (Psychological manipulation) Orchestrating the six apparitions required controlling the perceptual experiences of three percipients without relying on stimuli perceptible to bystanders, and without unintended side-effects. To achieve this by natural means, an agent would have to find a way to transmit a large amount of information into the percipients’ nervous systems through an unknown, imperceptible channel, then hijack their perception by hacking their chaotic, individualized neural codes, all without disturbing ordinary functioning.

• (Atmospheric manipulation) The Miracle of the Sun required exercising control over the atmosphere to create a dramatic, large-scale spectacle exactly when and where it was promised. To achieve this by natural means, one would need to disperse storm clouds, spawn two anomalous light sources, and then choreograph the sources’ motion and optics in the turbulent lower atmosphere. Each of those feats would have required Star Trek-level science and technology, massive energetic inputs, and a number of collateral effects that were not observed.

— Muse’s original article

Just prima facie that’s way less plausible than children acting in a way currently uncaptured by developmental research or — yes, just to steelman things: — in a way that seems contrary to current developmental research. Surely you see that.